A note on Lamarle formula in Minkowski $3$-space

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Ufuk Öztürk
Kazım İlarslan
Esra Betül Koç Öztürk
Emilija Nesovic

Abstract

The Lamarle formula is known as a simple relation between the Gaussian curvature and the distribution parameter of a non-developable ruled surface. In this paper, we obtain the Lamarle formula of a non-developable ruled surface with pseudo null base curve and null director vector field in Minkowski $3$-space. We also obtain the corresponding striction line and distribution parameter of such surface. We prove that there is no Lamarle formula when the director vector field is spacelike and its derivative is null, because the ruled surface in that case is a lightlike plane. Finally, we give some examples.

Article Details

How to Cite
Öztürk, U., İlarslan, K., Koç Öztürk, E. B., & Nesovic, E. (2018). A note on Lamarle formula in Minkowski $3$-space. Tamkang Journal of Mathematics, 49(4), 291–300. https://doi.org/10.5556/j.tkjm.49.2018.2653
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Papers
Author Biographies

Ufuk Öztürk, Çankırı Karatekin University

Department of Mathematics, Faculty of Science, University of Çankırı Karatekin, 18100 Çankırı, Turkey.

Kazım İlarslan, Kırıkkale University

Department of Mathematics, Faculty of Sciences and Art, University of Kırıkkale, 71450 Kırıkkale, Turkey.

Esra Betül Koç Öztürk

A.H.R.Mah. Çayyolu Cad. Ada Sitesi B blok No:18/12, 18100 Çankırı, Turkey.

Emilija Nesovic, University of Kragujevac

Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia.

References

F. Dillen and W. Kuhnel, Ruled Weingarten surfaces in Minkowski {$3$}-space, Manuscripta Math., 98(1999), 307--320

M.P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, New Jersey, 1976.

S. Ersoy and M. Tosun, Lamarle formula in 3-dimensional Lorentz space, Math. Commun., 16(2011), 593--607.

H. Liu and Y. Yuan, Pitch functions of ruled surfaces and B-scrolls in Minkowski $3$-space, J. Geom. Phys., 62(2012), 47--52.

H. Pottmann and J. Wallner,Computational Line Geometry, Springer-Verlag, Berlin, 2001.

]K. L. Duggal and D. H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific, Singapore, 2007.

B. O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, New York, 1983.

S. Izumiya and S. Chino, Lightlike developables in Minkowski 3-space, Demonstratio Math., XLIII(2010), 387--399.

J. Walrave, Curves and surfaces in Minkowski space, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)--Katholieke Universiteit Leuven (Belgium), 1995.

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